
where \(\mathcal {F}(T_{1}) \equiv \left(g_{*}(T_{1}) / 3.38\right)^{\frac{3}{4}}\left(g_{s}(T_{1}) / 3.93\right)^{-1}\)  is an \(\mathcal {O}(1)\)  factor and \(\rho _{\rm DM}\simeq 1.26\, {\mathrm {keV}}/{\mathrm {cm}^{3}}\)  from Planck 2018 data [1]}.
For the \(Z_{\cal N}\)  scenario under discussion, it follows from maZNLargeN that this ratio can be rewritten in the large \({\cal N}\)  limit as
\(\frac{\rho _{a,0}}{\rho _{\rm DM}} \simeq 0.48 \, {\cal N}^{3/8}{z^{{\cal N}/4}}\, \theta _{1}^2\, \left(\frac{f_a}{ 10^{13} \mathrm {GeV}}\right)^{3/2} \mathcal {F}(T_{1})\,.\) 
